It's complicated and if you are in this position, you may want to get some professional help or a good book.
There are different type of stock grants "unrestricted", "restricted" and "restricted units". They are all treated differently. Unrestricted is basically treated as regular income based on the fair market value of the stock at the time of grant. For a private company (that has stock) the fair market value is typically set by the board of directors a few times a year.
This can be potentially dangerous. You owe income tax on the stock grant, but you don't have an easy way to sell the stock and actually make some cash. Some people may chose NOT to accept the grant to avoid this problem.
A restricted grant has some "strings attached". Typically it's a vesting period, i.e. you have to be an employee for XXX years before the grant becomes yours. In this case, it creates a taxable event at the time of vesting. The same risk applies: if the stock isn't publicly (or otherwise) traded at the time of vesting, you may end up with a sizable tax bill, but no good way to turn your stock into liquid cash to pay the tax.
For restricted grant, you can also opt to file a 83(b). This means you agree to pay the income tax on the stock at the time of grant and not when it vests. This can make a lot of sense if the stock is still really cheap and you expect a steep increase over the vesting period.
The takeaway here is that you may have to pay taxes on gains that you haven't made yet with assets that you can't cash. If you are in AMT territory, things may get more complicated still. No, this does not make a lot of sense, but, yes, that's the way it is. Welcome to the US Tax code.